Department of Economics
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WORKING PAPERS IN ECONOMICS
No 555
State-Variable Public Goods and
Social Comparisons over Time
Thomas Aronsson and Olof Johansson-Stenman
February 2013
ISSN 1403-2473 (print)
ISSN 1403-2465 (online)
State-Variable Public Goods and
Social Comparisons over Time
Thomas Aronsson* and Olof Johansson-Stenman
Abstract The optimal provision of a state-variable public good, where the global climate is the prime example, is analyzed in a model where people care about their relative
consumption. We consider both keeping-up-with-the-Joneses preferences (where people compare their own current consumption with others’ current consumption) and catching-up-with-the-Joneses preferences (where people compare their own current consumption with others’ past consumption) in an economy with two productivity types, overlapping generations and optimal nonlinear income taxation. The extent to which the conventional rules for public provision ought to be modified is shown to depend on the strength of such relative concerns, but also on the preference elicitation format.
Keywords: State variable public goods, asymmetric information, relative consumption, status, positional preferences, climate policy.
JEL Classification: D62, H21, H23, H41
A previous version of this paper was circulated under the title “State-Variable Public Goods
When Relative Consumption Matters: A Dynamic Optimal Taxation Approach”
Acknowledgement: The authors are grateful for constructive comments from seminars participants at Lund University, Stockholm School of Economics, Uppsala University and University of Arizona. They would also like to thank the Bank of Sweden Tercentenary Foundation, the Swedish Council for Working Life and Social Research, the Swedish Research Council FORMAS, the Swedish Tax Agency and the European Union Seventh Framework Programme (FP7-2007-2013) under grant agreement 266992 for generous research grants.
*
Address: Department of Economics, Umeå School of Business and Economics, Umeå University, SE – 901 87 Umeå, Sweden. Telephone: +46907865017. E-mail: Thomas.Aronsson@econ.umu.se
Address: Department of Economics, School of Business, Economics and Law, University of Gothenburg, SE – 405 30 Göteborg, Sweden. Telephone: +46317862538. E-mail: Olof.Johansson@economics.gu.se
1. Introduction
The present paper concerns the optimal provision of a state-variable public good, such that the public good can be seen as a stock that accumulates over time, in a dynamic economy where people have positional preferences for private consumption. The latter means that people derive utility from their own private consumption relative to that of other people, for which there is now much evidence both from questionnaire-based experiments and happiness research.1
The problem of characterizing the optimal provision of public goods under relative consumption concerns is not new in itself: it has been analyzed in a static setting with lump-sum taxes by Ng (1987) and Brekke and Howarth (2002), and in a static setting with second best taxation by Wendner and Goulder (2008), Aronsson and Johansson-Stenman (2008, forthcoming a) and Wendner (forthcoming). As a consequence of using static models, all these studies have focused on cases where the public good is a flow variable, and where the concept of relative consumption lacks a time-dimension. To our knowledge, the optimal provision of public goods under relative consumption concerns has not been analyzed before in a dynamic context. Since this dynamic problem for obvious reasons is more complex than its static counterpart examined in previous research, one may wonder whether the value added in terms of insights from such a study is worth the costs in terms of additional complexity. We argue that it is for at least two reasons.
First, most public goods share important state-variable characteristics, in the sense that their quality depend on previous actions, where the global climate stands out as a prime example. Indeed, the question concerning how much we should invest to combat climate change is one of the most important and discussed of our times. The current quality, and characteristics more generally, of the atmosphere are clearly not only affected by the actions taken today (such as current public abatement activities); they are also strongly affected by actions taken in previous periods (cf. Stern 2007). Correspondingly, actions taken today will affect the atmosphere for a long time. Similar arguments apply to many other public goods as well,
1
See, e.g., Easterlin (2001), Johansson-Stenman et al. (2002), Blanchflower and Oswald (2004), Ferrer-i-Carbonell (2005), Luttmer (2005), Solnick and Hemenway (2005), Carlsson et al. (2007), Clark and Senik (2010) and Corazzini, Esposito and Majorano (2012). See also evidence from brain science (Fliessbach et al. 2007; Dohmen et al. 2011).
including infrastructural investments such as roads, schools and hospitals. Therefore, it is clearly of high policy relevance to identify optimal provision rules for state variable public goods, as well as understand how these rules are modified due to relative consumption concerns, not least due to the important policy implications of such concerns found in other literature on taxation and public expenditure. Given the dynamic nature of the policy problem analyzed below, the paper will also, implicitly, touches on the discounting problem; yet, this
is not the main task, and we will not discuss any intergenerational equity issues.2
Second, the dynamic framework allows us to examine the implications of a broader set of measures of relative consumption by considering comparisons based on keeping-up–with-the-Joneses preferences, where each individual compares his/her current consumption with other people’s current consumption, simultaneously with intertemporal consumption comparisons. Such intertemporal comparisons may include comparisons with one’s own past consumption as well as comparisons with other people’s past consumption, where the latter will be referred
to as catching-up–with-the-Joneses.3 The extension to intertemporal social comparisons is
important for several reasons: (i) There is empirical evidence suggesting that people make
comparisons with their own past consumption and with the past consumption of others.4 (ii)
Intertemporal social comparisons have been found to have important implications for optimal
tax policy (Ljungqvist and Uhlig, 2000; Aronsson and Johansson-Stenman, 2012),5 and are
also consistent with the equity premium puzzle discussed in the literature on dynamic macroeconomics (e.g., Abel, 1990; Campbell and Cochrane, 1999). (iii) Finally, such
2
The choice of discount rate is perhaps the most discussed issue in the economics of global warming; see, e.g., Nordhaus (2007) and Stern (2007).
3
Other literature sometimes uses the concepts of “keeping-up-with-the-Joneses” and “catching-up-with-the-Joneses” in a more specific way based on assumptions about how the individual reacts to changes in others’ current and previous consumption; see, e.g., Alvarez-Cuadrado et al. (2004) and Wendner (2010 a, b).
4
See, e.g., Loewenstein and Sicherman (1991) and Frank and Hutchens (1993) for evidence suggesting that people make comparisons with their own past consumption. Senik (2009) presents further empirical evidence pointing at the importance of historical benchmarks. She finds that the individual well-being increases if the standard of living of the response person’s household increases by comparison to an internal benchmark given by the household’s standard of living 15 year ago, and if the individual has done better in life than his/her parents, ceteris paribus.
5
Other literature on optimal taxation under relative consumption concerns typically focus on atemporal (keeping-up-with-the-Joneses) comparisons; see, e.g., Boskin and Sheshinski (1978), Oswald (1983), Ireland (2001), Dupor and Liu (2003) and Kanbur and Tuomala (2010).
comparisons are also in line with recent research based on evolutionary models, such as Rayo and Becker (2007), in which there are evolutionary reasons for why people should compare their own current consumption with three distinct reference points: others’ current consumption, their own past consumption and others’ past consumption. Our study relates to Rayo and Becker in the sense that we consider all three comparisons simultaneously in terms of their implications for public good provision. In addition to the value of identifying how the optimal public good provision rule should be modified due to these extensions, we argue that it is equally important to identify the extent to which the basic insights from static models of public good provision under relative consumption concerns carry over to the dynamic case with state variable public goods. While some of the results derived below are similar to those found in static models, other results are distinctly different.
Policy rules for public goods depend on the set of tax instruments that the government has at its disposal. The model in the present paper builds on the models developed in Aronsson and Johansson-Stenman (2010, 2012), which address optimal income taxation under asymmetric information in an Overlapping Generations (OLG) framework with two ability-types but do
not consider public goods, the concern of the present paper.6 Such a framework gives a
reasonably realistic description of the information constraints inherent in redistribution policy; it also allows us to capture redistributive and corrective aspects of public good provision, as well as interaction effects between them, in a relatively simple way. Section 2 presents the OLG framework, preference structure and individual optimization problems, while Section 3 considers the optimization problems of firms. In Section 4, we describe the corresponding optimization problem facing the government. Section 5 presents rather general expressions for the optimal provision rule, which are valid for all kinds of social comparisons. Yet, while these results provide general insights on the incentives for public provision under relative consumption concerns, they are not directly interpretable in terms of the strength of such concerns. Therefore, the provision rules derived in the following sections 6 and 7 are expressed directly in terms of the degrees of positionality.
6
The seminal paper on public good provision under optimal nonlinear income taxation is Hylland and Zeckhauser (1979), whereas Boadway and Keen (1993) was the first study dealing with this problem based on the self-selection approach to optimal taxation developed by Stern (1982) and Stiglitz (1982).
Section 6 concerns the case where the individual only compares his/her own current consumption with other people’s current consumption (Keeping-up-with-the-Joneses preferences); as such, it builds on the model by Aronsson and Johansson-Stenman (2010) and extends it to encompass public goods. The results here are shown to depend crucially on the preference elicitation format. If people’s marginal willingness to pay for the public good is measured independently, i.e. without considering that other people also have to pay for increased public provision, then relative consumption concerns typically (for reasonable parameter values) work in the direction of increasing the optimal provision of the public good. However, this is not the case when a referendum format is used, so that people are asked for their marginal willingness to pay conditional on that all people will have to pay for increased public provision. Conditions are also presented for when a dynamic analogue of the conventional Samuelson (1954) rule applies.
Section 7 considers the more general case with both keeping-up and catching-up-with-the-Joneses preferences simultaneously, i.e. where people derive utility from their own consumption relative to the current and past consumption of others, as well as relative to their own past consumption; as such, the model extends the one in Aronsson and Johansson-Stenman (2012) to encompass public goods. Under some further simplifying assumptions (e.g., about the population size and how the concerns for relative consumption change over time), it is shown that the policy rule for public provision can be written as a straightforward extension of the corresponding policy rule derived solely on the basis of keeping-up-with-the-Joneses preferences in Section 6. However, contrary to the findings in Section 6, we also show that if individuals compare their own consumption with other people’s past consumption, a referendum format for measuring marginal benefits conditional on that others also have to pay for the public good no longer leads to a straight forward dynamic analogue to the Samuelson condition. Section 8 provides some concluding remarks. Proofs of all propositions (along with some other calculations) are provided in the Appendix.
2. The OLG Economy and Individual Preferences
We consider an economy where individuals live for two periods. An individual of generation t is young in period t and old in period t+1. We assume that each individual works during the first period of life and does not work during the second. Individuals differ in ability (productivity) and we simplify by considering a framework with two ability-types, where the
low-ability type (type 1) is less productive than the high-ability type (type 2). Each individual of ability type i and generation t cares about his/her consumption when young and when old,
i t
c and xti1; his/her leisure when young, zti; and the amount of the public good available when young and when old, Gt and Gt1. The individual also derives utility through his/her
relative consumption by comparison with (a) other people’s current consumption, (b) his/her own past consumption, and (c) other people’s past consumption. This is soon to be explained more thoroughly.
The life-time utility function faced by ability-type i of generation t is written as follows:
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ( , , , , , , , , , ) ( , , , , , , , , ) ( , , , , , , , ) i i i i i i i i i i i t t t t t t t t t t t t t t t t t i i i i i i i i t t t t t t t t t t t t t t i i i i t t t t t t t t t U V c z x c c x c x c c c x c G G v c z x c c x c c c x c G G u c z x c c c G G , (1) where t t ti i ti1 ti / ti ti 1 i i
c
n c n x
n n denotes the average consumption in the economyas a whole in period t; nti measures the number of young individuals of ability-type i in period t, (implying, of course, that nti1 represents the number of old individuals of ability-type i in period t). The five consumption differences in equation (1) – as represented by the fourth to eights argument in the function Vi( ) - are measures of relative consumption, and imply that the individual compares his/her current consumption with (a) the current average consumption when young and when old, i.e., ctict and xti1ct1; (b) his/her own
consumption one period earlier when old, i.e., i1 i
t t
x c ; and (c) the average consumption one
period earlier when young and when old, i.e., ctict1 and xti1ct. Two things are worth noticing. First, the relative consumption is defined as the difference between the individual’s own consumption and the appropriate reference measure; this approach is technically convenient and also taken in many previous studies (e.g., Akerlof, 1997; Corneo and Jeanne,
1997; Ljungqvist and Uhlig, 2000; Bowles and Park, 2005; Carlsson et al.; 2007).7 Second,
7
An alternative is the (slightly less technically convenient) ratio comparison, where the individual’s relative consumption is defined by the ratio between the individual’s own consumption and the reference measure (e.g., Boskin and Sheshinski, 1978; Layard, 1980; Abel, 2005; Wendner and Goulder, 2008). Aronsson and Johansson-Stenman (forthcoming b) derived optimal income taxation rules in a static model based on both difference and ratio comparisons and concluded that the main qualitative insights obtained are unaffected by
the measures of reference consumption implicit in comparisons (a) and (c) are indicators of the average consumption in the economy as a whole, which is the common way to define
reference consumption in earlier studies.8
To explain the second utility formulation in equation (1), i.e. the function i( )
t
v , note that i
t
c
and xti1 are decision variables of the individual; therefore, we can without loss of generality
use the simpler function vti( ) on the second line, where the effect of xti1cti on utility is
embedded in the effects of i
t
c and i1
t
x . As such, habit formation does not produce a
corrective motive for provision of public goods. The function uti( ) is a convenient reduced
form to be used in some of the calculations presented below; however, uti( ) also represents
the most general utility formulation in the sense of not specifying how the relative consumption comparisons are made (other than that other’s consumption gives rise to negative externalities).
The policy rules for public provision examined below reflect the extent to which relative consumption concerns are important for individual well-being. As such, it is useful to measure the degree to which such concerns matter for each individual, which we will do by employing the second utility formulation, vti( ) , in equation (1). By using the following variables: , i c i t ct ct , , 1 1 1 i x i t xt ct , , 1 i c i t ct ct , , 1 1 i x i t xt ct
as short notations for the four differences in the function vti( ) , we follow Aronsson and
Johansson-Stenman (2012) and define the degree of current consumption positionality for
ability-type i of generation t when young and old, respectively, as
comparison type. The same applies here: although the choice to focus on difference comparisons instead of ratio comparisons will affect the exact form of the policy rules derived below, it is of no significance for the qualitative insights from our analysis.
8
Almost all previous studies on optimal tax and/or expenditure policy under relative consumption concerns assume that individuals compare their own consumption with a measure of average consumption. In a study of optimal taxation, Aronsson and Johansson-Stenman (2010) consider alternative reference measures based on within-generation and upward comparisons, respectively, and find tax policy responses that are qualitatively similar to those that follow if the reference point is based solely on the average consumption. Upward comparisons are also analyzed by Micheletto (2011).
, , , , , c c c i t i c t i i i t c t t v v v v , (2) , , 1 , , , x x x i t i x t i i i t x t t v v v v , (3)
while we define the degree of intertemporal consumption positionality when young and when old, respectively, as follows:
, , , , , c c c i t i c t i i i t c t t v v v v , (4) , , 1 , , , x x x i t i x t i i i t x t t v v v v . (5)
Here, sub-indexes indicate partial derivative, i.e. vt ci, vti( ) / cti, vt xi, vti( ) / xti1,
, , c ( ) / i i i c t t t v v , , x ( ) / ,1 i i i x t t t v v , , c ( ) / , i i i c t t t v v , and , x ( ) / ,1 i i i x t t t v v .
Note that equations (2) and (3) reflect comparisons with other people’s current consumption,
i.e. the keeping-up-with-the-Joneses motive for relative consumption. The variable ti c, ,
which denotes the degree of current consumption positionality when young, is interpretable as the fraction of the overall utility increase from an additional dollar spent on private consumption when young in period t that is due to the increased consumption relative to the
average consumption in period t. For example, if ti c, 0.3 then 30% of the utility increase
from the last dollar spent by the individual when young in period t is due to the increased relative consumption compared to other people’s current consumption in the same period; hence, 70% is due to a combination of increased absolute consumption and increased relative
consumption compared to other people’s past consumption. The variable ,
1 i x t
has an
analogous interpretation for the old individual in period t+1. Equations (4) and (5) reflect comparisons with other people’s past consumption, i.e. the catching-up-with-the-Joneses
an additional dollar spent when young in period t that is due to the increased consumption
relative to other people’s past consumption; ,
1 i x t
has an analogous interpretation for the old
consumer in period t+1.
Returning finally to equation (1), the state variable public good is governed by the difference equation
1
(1 )
t t t
G g G , (6)
where gt is the addition to the public good in period t, provided by the government, and is
the rate of depreciation. Therefore, the traditional flow-variable public good appears as the
special case where 1.
Each individual of any generation t treats the measures of reference consumption, i.e. ct1, ct
and ct1, as exogenous during optimization (while these measures are of course endogenous
to the government, as will be explained below). Let lti denote the hours of work by an individual of ability-type i in period t, while wti denotes the before-tax wage rate, sti savings, and rt the market interest rate in period t. Also, let Tt( ) and t1( ) denote the payments of
labor income and capital income taxes in period t and t+1, respectively. The individual
budget constraint can then be written as
( ) i i i i i i t t t t t t t w l T w l s c , (7) 1 1 1 1 (1 ) ( ) i i i t t t t t t s r s r x . (8)
The individual first order conditions for work hours and savings are standard. Since these conditions are not used to derive the cost benefit rules for public goods analyzed below, they are presented in the Appendix.
We model the production sector in a standard way: it consists of identical competitive firms, whose number is normalized to one for notational convenience, producing a homogenous good under constant returns to scale. This output is used for both public and private consumption, as described in section 4. The production function is written as
1 2
( , , )
t t t t
Y F L L K , (9)
where Yt denotes the output (national product), while
i i i t t t
L n l is the total number of hours of
work supplied by ability-type i in period t, and Kt is the capital stock in period t.The representative firm obeys the standard optimality conditions
1 2 ( , , ) i i t t t t L F L L K w for i=1, 2, (10) 1 2 ( , , ) K t t t t F L L K r. (11)
To simplify the calculations below, we introduce an additional assumption; namely, that the
relative wage rate (often called wage ratio) in period t, 1 2
1 2
/ /
t w wt t FL FL
does not depend
directly on Kt, which holds for standard constant returns to scale production functions such
as the Cobb-Douglas and CES. This means that the intertemporal tradeoff faced by the government will be driven solely by the interest rate.
4. The Government’s Optimization Problem
We will use an as general social welfare function as possible, by assuming that social welfare increases with the utility of any individual type alive at any time period, without saying anything more regarding these relationships. Following Aronsson and Johansson-Stenman (2010, 2012), we then assume that the government faces a general social welfare function as follows: 1 1 2 2 1 1 2 2 0 0 0 0 1 1 1 1 ( , , , ,....) W W n U n U n U n U , (12)
which is increasing in each argument. Since ability is assumed to be private information, the public policy must also satisfy a self-selection constraint. As in most of the literature on
redistribution under asymmetric information, we consider a case where the government wants to redistribute from the high-ability to the low-ability type, implying that the self-selection constraint must prevent the high-ability type from acting as a mimicker, i.e.
2 2 2 2 2 1 1 1 1 2 1 1 1 2 1 1 1 1 ( , , , , , , , ) ˆ ( ,1 , , , , , , ) t t t t t t t t t t t t t t t t t t t t t U u c z x c c c G G u c l x c c c G G U . (13)
The left hand side of equation (13) denotes the utility of the high-ability type, while the right hand side is the utility of the mimicker (a high ability type who pretends to be a low-ability type); the time endowment available for work hours and leisure is normalized to one. The variable t tl1 is interpretable as the mimicker’s labor supply; since t wt1/wt2 1, we have
1 1 t tl lt
. Notice also that equation (13) is based on the assumption that all income is
observable to the government: therefore, a high-ability mimicker must actually mimic the point chosen by the low-ability type on both tax function and, therefore, consume the same amount as the low-ability type in both periods.
The resource constraint for this economy means that the output is used for private consumption as well as private and public investment, and is written as follows:
2 1 2 1 1 1 ( , , ) i i i i 0 t t t t t t t t t t i F L L K K n c n x K g
. (14)The second best problem will be formulated as a direct decision problem, i.e. to choose lt1,
1 t
c , x1t, lt2, ct2, xt2, Kt, gt and Gt for all t to maximize the social welfare function presented in equation (12) subject to equations (6), (13) and (14). The government also recognizes that the measures of reference consumption are endogenous as defined by the mean-value formula presented in Section 2. The Lagrangean can be written as
1 1 2 2 1 1 2 2 2 2 0 0 0 0 1 1 1 1 2 1 2 1 1 1 1 ˆ ( , , , ,....) ( , , ) [ ] (1 ) t t t t i i i i t t t t t t t t t t t t i t t t t t W n U n U n U n U U U F L L K K n c n x K g g G G
, (15)where , and are Lagrange multipliers. The first-order conditions are presented in the
Appendix. These conditions will now be used to analyze the optimal provision of the public good.
5. A General Rule for Public Good Provision when Relative Consumption Matters
In this section we present general optimality conditions for the public good provision in a format that facilitates straightforward economic interpretations and comparisons with the benchmark case with no relative consumption concerns. More specifically, the optimal provision rules will be expressed in terms of what we will dente the positionality effect, i.e. the welfare effect associated with changed reference consumption per se.
Let uˆt2 u ct2( ,11t l xt1, t11,ct1, ,c ct t1,G Gt, t1) denote the utility faced by the mimicker of
generation t based on the function ut2( ) in equation (1). We can then define the marginal rate
of substitution between the public good and private consumption for the young and old ability-type i, and for the young and old mimicker, in period t as follows:
, , , , t i t G i t G c i t c u MRS u , ,, 1, 1, t i t G i t G x i t x u MRS u , 2 , 2, , 2 , ˆ ˆ ˆ t t G t G c t c u MRS u and 2 1, 2, , 2 1, ˆ ˆ t t Gt G x t x u MRS u .
To shorten the formulas to be derived, we shall also use the short notations
, , , , 1 , i i t i i t t G t G c t G x i i MB
n MRS
n MRS (16) 2 1, 2, 2 1, 2, , , ˆ , 1 1, , ˆ , ˆ t t ˆ t t t tut cMRSG c MRSG c tut xMRSG x MRSG x (17)for the sum of the marginal willingness to pay for the public good (measured as the marginal rate of substitution between the public good and private consumption) among those alive in period t and the difference in the marginal value attached to the public good between
low-ability-type and the mimicker (measured both for the young and old) in period t, respectively.
To facilitate later interpretations, we assume that MBt G, is decreasing in Gt.9 We are nowable
to derive the following result:
Proposition 1. The optimal provision of the public good is characterized by
, , 0 1 1 t G t t G t t t t t MB MB N c
. (18)Following Aronsson and Johansson-Stenman (2010), the partial derivative of the Lagrangean
with respect to the reference consumption in period t, i.e. /ct, will be called the
positionality effect in period t, and reflects the overall welfare consequences of an increase in
t
c , holding each individual’s own consumption constant. As such, it is a measure of the
“positional externality” of private consumption. While it is reasonable to expect /ct to
be negative, since for each individual , 0
t i t c u and 1 ,t 0 i t c
u , it is theoretically possible that it
is positive due to effects through the self-selection constraint to be discussed more thoroughly in the following sections.
Before interpreting Proposition 1 further, let us first consider the special case where 1, in
which the state-variable public good is equivalent to an atemporal control (or flow) variable,
i.e. Gt gt, so that we can simplify equation (18) and obtain:
Corollary 1. If the public good is a flow variable, so that 1, then the optimal provision of the public good satisfies
, , 1 t G t G t t t t MB MB N c . (19)
Equation (19) is analogous to the formula for public provision derived in a static model by Aronsson and Johansson-Stenman (2008). The right-hand side is the direct marginal cost of
9
A sufficient – yet not necessary – condition for this property to hold is that private and public consumption, if measured in the same period, are weak complements in the utility function.
providing the public good, which is measured as the marginal rate of transformation between the public good and the private consumption good and is normalized to one, whereas the left-hand side is interpretable as the marginal benefit of the public good adjusted for the influences of the self-selection constraint and positional preferences, respectively. With a flow-variable public good, the main differences between a static model and the intertemporal model analyzed here are that the self-selection effect and positionality effect relevant for public provision in period t reflect the incentives facing generations t and t-1, as the high-ability type in each of these generations may act as a mimicker in period t.
Let us now return to the case with a state variable public good, i.e. where 1. Equation (18) essentially combines the policy rule for a state-variable public good in an OLG model without positional preferences derived by Pirttilä and Tuomala (2001) with an indicator of how the marginal benefit of an incremental public good is modified by relative consumption concerns. Again, the right-hand side is the direct marginal cost of a small increase in the
contribution to the public good in period t, which is measured as the marginal rate of
transformation between the public good and the private consumption good, whereas the left-hand side measures the marginal benefit of an increase in the contribution to the public good
in period t adjusted for the influences of the self-selection constraint and positional
preferences, respectively. Note that this measure of adjusted marginal benefit is intertemporal
as an increase ingt, ceteris paribus, affects the utility of each ability-type, as well as the
self-selection constraint and the welfare the government attaches to increased reference consumption, in all future periods. The latter means that the marginal benefit of an increment to the public good in period t depends an intertemporal sum of positionality effects; not just the positionality effect in period t. Therefore, positional concerns affect state variable and flow variable public goods differently, which will be described more thoroughly below.
6. Optimal Provision Rules with Keeping-up-with-the-Joneses Preferences
The positionality effect included in equation (18) is crucial for our understanding of how the incentives underlying public provision depend on the relative consumption concerns. In this section, we assume that the positional preferences are of the keeping-up-with-the-Joneses type, meaning that each individual derives utility from his/her own current consumption relative to the current average consumption in the in the economy as a whole, and that each
individual makes this comparison both when young and when old. As indicated above, we abstract from catching-up-with-the-Joneses comparisons here; such comparisons are addressed in Section 8 below. This simplification means that the variables ti c, cti ct1
, and
,
1 1
i x i t xt ct
vanish from equation (1) and, as a consequence, that the intertemporal degrees
of positionality are equal to zero.
When the preferences are of the keeping-up-with-the-Joneses type, the positionality effect only reflects current degrees of positionality. By using equations (2) and (3) – which represent measures of the current degree of positionality at the individual level – we can define the average degree of current consumption positionality in period t as follows:
, 1 , (0,1) i i i x t i c t t t t i t i t n n N N
, (20a)where Nt
i[nti1nti] denotes total population in period t . We also introduce an indicator of the difference in the degree of current consumption positionality between the mimicker and the low-ability type in period t, td, such that2 2 1ˆ , ˆ2, 1, ˆ, ˆ2, 1, t t t x t t c d x x c c t t t t t t t t t u u N N , (20b)
where the symbol “^” denotes “mimicker” (as before), while the superindex “d” stands for
“difference.” Thus, d
t
reflects an aggregate measure of the positionality differences between
the young mimicker and the young low-ability type and between the old mimicker and the
old low-ability type, respectively. Consequently, td 0 (0) if the mimicker is always, i.e.
both when young and old, more (less) positional than the low-ability type. Following Aronsson and Johansson-Stenman (2010), the positionality effect associated with the keeping-up-with-the-Joneses type of positional preferences can then be written as follows;
1 d t t t t t t N c . (21)
Therefore, the overall welfare effects of an increase in the level of reference consumption in period t, ceteris paribus, contains two components. The first is the average degree of current positionality, t, which contributes to decrease the right hand side of equation (21). This
negative effect arises because the utility facing each individual of generation t depends
negatively on ct via the argument i t t
c c in the utility function, and the utility facing each
individual of generation t-1 depends negatively on ct via the argument xtict. As such, the
average degree of current positionality reflects the magnitude of the positional externality.
The second component in equation (21), td, appears because the mimicker and the
(mimicked) low-ability type typically differs with respect to the degree of positionality, which the government may exploit to relax the self-selection constraint. If td 0 (0),
increased reference consumption in period t leads to a relaxation (tightening) of the
self-selection constraint, as it means that the mimicker is hurt more (less) than the low-ability
type. As a consequence, this effect may either counteract ( d 0
t
) or reinforce ( d 0
t
) the
negative positional consumption externality.
By substituting equation (21) into equation (19), we can derive the following result:
Proposition 2. The optimal provision of the public good based on keeping-up-with-the-Joneses preferences is characterized by
, 0 1 1 1 1 d t t t G t t t MB
. (22)The interesting aspect of Proposition 2 is that the effects of positional concerns are captured by a single multiplier, (1td) / (1t), which is interpretable as the
“positionality-weight” in period t. The average degree of positionality, t, contributes to scale up the
aggregate instantaneous marginal benefit and, therefore, increases the provision of the public good. As explained above, the effect of td (the measure of differences in the degree of positionality between the mimicker and the low-ability type) can be either positive or
negative. If td 0, this mechanism contributes to scale down the marginal benefit of public
private consumption leads to a relaxation of the self-selection constraint in this case (as the mimicker is more positional than the low-ability type). If td 0, on the other hand, this mechanism works in this opposite direction.,
Therefore, a sufficient (not necessary) condition for the positionality weight in period t to
scale up the aggregate instantaneous marginal benefit of the public good in that period is that 0
d t
, meaning that the low-ability type is at least as positional as the mimicker. In the Appendix, we derive the following result more generally:
Proposition 3. A neccessary and sufficient condition for the joint impact of present and future positionality effects to increase the contribution to the public good in period t is that
, 0 1 0 1 d t t t G t MB
.Hence, a sufficient condition is that the low-ability type is predominantly at least as positional as the mimicker in the sense that
, 0 1 0 1 d t t G t MB
.Note that even though the second condition in Proposition 3 is much stronger than the first, it still does not require the low-ability types to be at least as positional as the mimickers in all periods. Instead, as long as the low-ability type is predominantly as least as positional as the mimicker – which imposes a condition on a weighted average of future differences in the degree of positionality – this is perfectly consistent with the possibility that the mimicker is more positional than the low-ability type during certain periods or intervals of time.
Let us next consider conditions for when the second-best adjustments through the impacts on the self-selection constraints, i.e. the effects of the variables t and td for all t, vanish from the policy rule for public provision. We have derived the following result;
Proposition 4. If leisure is weakly separable from private and public consumption in the sense that the utility function can be written as Uti q h c xti( ( ,t ti ti1, i ct, , ti x,1,G Gt, t1), )zti for all t, then the optimal policy rule for the public good is given by
, 0 1 1 1 t G t t t MB
. (23)Note that while we still allow for type-specific preferences, here the function ht( ) is common to all consumers of generation t. In the absence of relative consumption concerns (in which
case t 0 for all t), equation (23) coincides with a dynamic analogue to the standard
Samuelson condition. This result is modified here because the policy rule still reflects a desire to correct for positional externalities, which works to increase the marginal benefit of
the public good (since 1/(1t) 1 for all t by assumption). The intuition behind Proposition
4 is that if leisure is weakly separable from the other goods in the utility function – and with the additional restriction that the function ht( ) is common for the two ability-types – it
follows that t td 0 for all t, i.e. neither the marginal willingness to pay for the public
good nor the degree of positionality differs between the mimicker and the low-ability type. As a consequence, there is no longer an incentive for the government to modify the provision of the public good to relax the self-selection constraint.
Following Aronsson and Johansson-Stenman (2008), it is interesting to consider the role of preference elicitation for the public good. Note first that individual benefits of the public good are so far measured by each individual’s marginal willingness to pay for a small
increment, ceteris paribus, i.e. while holding everything else, including others’ private
consumption, fixed. At the same time, increased public provision typically comes together with other changes, notably that one’s own as well as other people’s taxes or charges are increased. In one frequently used method, the contingent valuation method, it is typically recommended (see Arrow et al. 1993) that a realistic payment vehicle is used when asking people about their maximum willingness to pay. One commonly used payment vehicle is to ask subjects how they would vote in a referendum where everybody would have to pay a certain amount, the same for all, through increased taxes (or charges) for the improvement. In the standard case where people do not care about relative consumption, this formulation has no important theoretical implication. Here, however, it does. To see this, let us define the marginal rate of substitution between the public good and private consumption at any time, t, conditional on the requirement that ctict and xti1ct1 remain constant, which would
follow if the willingness to pay question were supplemented by the information that everybody has to pay the same amount for an incremental public good.
With reference to equation (1), this measure of marginal willingness to pay, conditional on that others would have to pay equally much on the margin, can then be defined as follows:
Definition. An individual’s conditional marginal willingness to pay for the public good when
young and old, respectively, is defined by:
, , , , t i t G i t G c i t c v CMRS v , (24a) 1, , , 1, t i t G i t G x i t x v CMRS v . (24b)
In a way similar to equation (17), we may construct an aggregate marginal benefit measure
consisting of the sum of all people’s (alive in period t) marginal willingness to pay for the
public good, conditional on that others will also have to pay equally much on the margin, as follows: , , , , 1 , i i t i i t t G t G c t G x i i CMB
n CMRS
n CMRS . (25)The question is then how the optimal provision rule will change if expressed as a function of
, t G CMB instead of MBt G, ? By using , , 1 cov , 1 t G c t t t t G c CMRS CMRS
as a short notation for the (normalized) covariance between the degree of non-positionality,
measured by 1ti, and the marginal willingness to pay for the public good, we have:
Proposition 5. Based on keeping-up-with-the-Joneses preferences, and expressed in terms of conditional marginal WTPs, the optimal policy rule for public good provision is given by
, 0 1 1 d 1 1 t t G t t t t CMB
. (26)Compared to equation (22), we can observe two differences (in addition to the replacement of
, t G
MB by CMBt,G): First and foremost, the benefit-amplifying factor 1/ (1t) is not part of equation (26). The intuition is straightforward. If others’ consumption is held constant, each individual’s willingness to pay for increased public good provision would be reduced by the fact that his/her relative consumption decreases. However, if each individual’s relative consumption is held constant, as in Proposition 5, there is obviously no such effect. Second,
equation (26) include a factor
1 t
, for which the intuition can be given as follows. If theconditional marginal WTP differs between the types, and all people will have to pay the same amount on the margin, then those with a higher conditional marginal WTP will obtain a utility increase, while the others will face a utility loss. The utility increase, in monetary terms, will more than outweigh the utility loss if and only if those with a higher conditional marginal WTP are less positional, i.e. if and only if the covariance between the degree of non-positionality and the conditional marginal WTP is positive.
It is finally interesting to analyze whether there is some special case in which the second-best policy rule for the public good reduces to a first-best policy rule. It turns out that there is, and the following result gives sufficient conditions under which an intertemporal analogue to the Samuelson rule applies:
Proposition 6. If - in addition to the conditions underlying Proposition 4 - we assume that (i) the degree of positionality is the same for both ability-types in all periods, both when young and when old, and (ii) the market interest rate is constant over time, then the optimal provision of the public good, expressed in terms of conditional marginal WTPs, is given by
, 0 1 1 (1 ) t G CMB r
. (27)Given that the degree of current consumption positionality is the same for everybody (in which case there is no correlation between the marginal willingness to pay and the degree of current consumption positionality at the individual level), then a weighted sum over time of
instantaneous marginal benefits should equal the marginal cost of an incremental public good. In other words, an intertemporal analogue to the traditional Samuelson rule applies. What is less clear, perhaps, is how we should apply this or any other policy rule in practice, as we would need information about the willingness to pay for the public good by future generations. However, before taking the discussion about implementation to any greater detail, it is important to know the point of departure.
Note also that in the special case where 1, i.e. the case where the public good is a flow
variable, Proposition 6 implies:
Corollary 2. In addition to the conditions governing Proposition 6 (except for a constant interest rate which is not needed here), suppose that the public good is a flow variable, so that 1. The policy rule for the public good then simplifies to read
, 1 t G
CMB . (28)
Corollary 1 thus implies that the conventional Samuelson (1954) rule, expressed in marginal willingness to pay conditional on the fact that also others have to pay on the margin, holds for each moment in time.
7. Optimal Provision of the Public Good under both keeping-up-with-the-Joneses and catching-up-with-the-Joneses preferences
The analysis carried out in earlier sections is based on the assumption that the only measure of reference consumption at the individual level, in any period, is based on the average consumption in that particular period. Although this idea accords well with earlier literature on public policy and positional preferences, it neglects the possibility that agents also compare their own current consumption with both their own past consumption and that of other people. In this section, we will present and analyze the more general model that takes all these comparisons into account.
Note once again that equation (18) holds generally, i.e. irrespective of which form the relative consumption concerns take. To be able to consider keeping-up-with-the-Joneses preferences
simultaneously with catching-up-with-the-Joneses preferences, we must explore the positionality effect, /ct, for this more general case. The positionality effect will then depend both on the current and intertemporal degrees of positionality. Therefore, in a way similar to the average degree of current consumption positionality in equation (20a), we use equations (4) and (5) to define the average degree of intertemporal consumption positionality as follows: , 1 , (0,1) i i i x t i c t t t t i t i t n n N N
. (29a)Furthermore, and by analogy to the variable td defined in equation (20b), which is a
summary measure of differences in the degree of current positionality between the mimicker and the low-ability type in period t, we define a corresponding measure of differences in the degree of intertermporal positionality between the mimicker and the low-ability type,
2 2 1ˆ , ˆ2, 1, ˆ, ˆ2, 1, t t t x t t c d x x c c t t t t t t t t t u u N N . (29b)
The variable td has the same general interpretation as td. In other words, td 0 (0) if the young and old mimicker in period t are more (less) positional than the corresponding low-ability type, where positionality is measured relative to other people’s past consumption. To simplify the notation and facilitate comparison with equation (21), we use the following short notation: 1 1 1 1 [ ] [ ] 1 1 d d t t t t t t t t t t t N N B .
We show in the Appendix (along with the proof of Proposition 7 below) that the positionality effect associated with this more general model can be written as
1 11 1 i t j t t i i j t t j B B c
. (30)The variable Bt in equation (30) is analogous to the right hand side of equation (21) with the modification that it also reflects the intertemporal (not just the current) degrees of positionality. As such, it contains two additional components. First, Nt1 t1 t1/(1t)0 is interpretable as the value of the positional externality associated with the catching-up-with-the-Joneses motive for relative consumption comparisons. The underlying mechanism is, of course, that ct directly affects individual utility negatively via the argument 1
i
t t
x c in the
utility function. Second, the component 1 1 1/(1 )
d
t t t t
N reflects the corresponding
welfare effects through the self-selection mechanism in period t+1. In a way similar to the analogous measure of differences in the current degree of positionality between the mimicker and the low-ability type, this effect means that increased reference consumption in period t may either contribute to relax (td10) or tighten (td10) of the self-selection constraint. The final component on the right hand side of equation (30) arises due to an intertemporal chain reaction: the intuition is that the catching-up-with-the-Joneses motive for consumption comparisons, i.e., that other people’s past consumption affects utility, means that the welfare effects of changes in the reference consumption are no longer time-separable (as they would be without intertemporal consumption comparisons).
By substituting equation (30) into equation (19), we can derive the following result:
Proposition 7. The optimal provision of the public good based on keeping.-up-with–the-Joneses and catching-up-with-the-keeping.-up-with–the-Joneses preferences is characterized by
, , 0 1 1 1 1 1 1 i t j t G t t G t t t i i j t t t t j MB MB B B N
. (31)The basic intuition behind Proposition 7 is analogous to that of Proposition 2; yet with the modification that the catching-up-with-the-Joneses motive for consumption comparisons is present in equation (31). This means that (i) increases in the average degrees of positionality (both in the current and intertemporal dimensions) typically contribute to increased provision of the public good, and (ii) differences in the degrees of positionality between the mimicker and the low-ability type contribute to increase (decrease) the optimal provision of the public
good if the low-ability type is predominantly more (less) positional than the mimicker is both dimensions.
At the same time, equation (31) is not very tractable, due to the intertemporal chain reaction caused by the catching-up-with-the-Joneses motive for relative consumption, and the interpretation of its different components is far from obvious. Yet, by making some additional simplifying assumptions, we are able to simplify the positionality effect given by equation (30) considerably, and also derive a policy rule for public provision that takes the same form as equation (22).
Assumptions A. t , t ,
d d
t
, td d, Nt N and rt r t
In other words, our measures of degrees of average positionality and positionality differences between the types of people as well as the population size and the interest rate are assumed to be constant over time. While these are of course important restrictions, they are hardly very strong assumptions, and similar assumptions are frequently made in the comparable
catching-up-with-the-Joneses literature.10 It should also be noted that the model is still general enough
to reflect different preferences between ability-types types, including different degrees of positionality.
Following Aronsson and Johansson-Stenman (2012), we can now define the average degree of total consumption positionality and the difference in the degree of total consumption positionality between the mimicker and the low-ability type, respectively, in present value terms as 1 r , 1 d d d r .
Therefore, the average degree of total consumption positionality is measured as the average degree of current consumption positionality plus the present value of the average degree of intertemporal consumption positionality; the reason for calculating the present value is, of course, that the intertemporal externality caused in period t gives rise to disutility in period
10
t+1. In a similar way, the measure of differences in the degree of total consumption positionality between the mimicker and the low-ability type, d , reflects differences in the
degree of current consumption positionality, d , and the (present value of) differences in
intertemporal positionality, d / (1r). We can then, under Assumptions A, show (see the
Appendix) that equation (29) reduces to read
1 d t t N c ,
which takes the same general form as in the absence of the catching-up-with-the-Joneses motive for relative consumption, i.e. as equation (21). This implies that we are able to present straightforward extensions of Propositions 2-6 and Corollary 2 to the more general case, where we also consider catching-up-with-the-Joneses preferences:
Proposition 2’. Under Assumptions A, and based on keeping.-up-with–the-Joneses and catching-up-with-the-Joneses preferences, the optimal provision of the public good is characterized by , 0 1 [1 ] 1 1 d t t G t t MB
. (32)Technically, the only difference compared to equation (22) is that the positionality-weight
based on current degrees, (1d) / (1), is here replaced by a corresponding
positionality-weight based on total degrees, (1d) / (1). The intuition is that comparisons with other
people’s past consumption imply that the (young and old) individuals alive today impose a negative positional externality on the individuals alive in the next period, i.e. the higher the consumption in period t, ceteris paribus, the greater will be the utility loss due to lower
relative consumption in period t+1. As such, this intertemporal externality must be
considered simultaneously with the (atemporal) externality that affects others today. Due to Assumptions A, the striking implication of Proposition 2’ is that the current and intertermporal aspects of consumption positionality affect the incentives for public good
provision in exactly the same way. Therefore, the following results for when positional
concerns lead to increased contributions to the public good are analogous to Proposition 3:
Proposition 3’. Under Assumptions A, and based on keeping.-up-with–the-Joneses and catching-up-with-the-Joneses preferences, a neccessary and sufficient condition for the joint impact of present and future positionality effects to increase the contribution to the public good in period t is that d 0. Hence, a sufficient condition is that the low-ability type is at least as positional as the mimicker in the sense that d 0.
Again, the result from Section 7 carries over with the only modifications that and dare
replaced by and d
, respectively. Let us then consider conditions for when the
second-best adjustments through the impacts on the self-selection constraints vanish from the policy rule for public provision. We can derive the following analogue to Proposition 4:
Proposition 4’. Given the conditions underlying Proposition 3’, and if leisure is weakly separable from private and public consumption in the sense that the utility function can be written as Uti q h c xti( ( ,t ti ti1, ti c, , ti x,1, ti c, , ti x,1,G Gt, t1), )zti for all t, then the optimal provision of the public good is characterized by
, 0 1 1 1 t G t t MB
. (33)Note that the separability condition in this case also includes the variables reflecting consumption comparisons over time. Consequently, the conditions for when the second-best adjustments through the impacts o
